guanjihuan/guanjihuan.com
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

update

Browse Source
This commit is contained in:
guanjihuan
2023-11-07 03:38:46 +08:00
parent 1e7c4c0e68
commit bc3890c25b
212 changed files with 0 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files

50
2020.01.05_calculation_of_Chern_number_by_definition_method/calculation_of_Chern_number_by_definition_method.m Normal file
View File

@@ -0,0 +1,50 @@
% This code is supported by the website: https://www.guanjihuan.com
% The newest version of this code is on the web page: https://www.guanjihuan.com/archives/3932
% 陈数定义法
clear;clc;
n=100; % 积分密度
delta=1e-9; % 求导的偏离量
C=0;
for kx=-pi:(1/n):pi
for ky=-pi:(1/n):pi
VV=get_vector(HH(kx,ky));
Vkx=get_vector(HH(kx+delta,ky)); % 略偏离kx的波函数
Vky=get_vector(HH(kx,ky+delta)); % 略偏离ky的波函数
Vkxky=get_vector(HH(kx+delta,ky+delta)); % 略偏离kx，ky的波函数
if sum((abs(Vkx-VV)))>0.01 % 为了波函数的连续性（这里的不连续只遇到符号问题，所以可以直接这么处理）
Vkx=-Vkx;
end
if sum((abs(Vky-VV)))>0.01
Vky=-Vky;
end
if sum(abs(Vkxky-VV))>0.01
Vkxky=-Vkxky;
end
% 价带的波函数的berry connection，求导后内积
Ax=VV'*(Vkx-VV)/delta; % Berry connection Ax
Ay=VV'*(Vky-VV)/delta; % Berry connection Ay
Ax_delta_ky=Vky'*(Vkxky-Vky)/delta; % 略偏离ky的berry connection Ax
Ay_delta_kx=Vkx'*(Vkxky-Vkx)/delta; % 略偏离kx的berry connection Ay
% berry curvature
F=((Ay_delta_kx-Ay)-(Ax_delta_ky-Ax))/delta;
% chern number
C=C+F*(1/n)^2;
end
end
C=C/(2*pi*1i)
function vector_new = get_vector(H)
[vector,eigenvalue] = eig(H);
[eigenvalue, index]=sort(diag(eigenvalue), 'descend');
vector_new = vector(:, index(2));
end
function H=HH(kx,ky)
H(1,2)=2*cos(kx)-1i*2*cos(ky);
H(2,1)=2*cos(kx)+1i*2*cos(ky);
H(1,1)=-1+2*0.5*sin(kx)+2*0.5*sin(ky)+2*cos(kx+ky);
H(2,2)=-(-1+2*0.5*sin(kx)+2*0.5*sin(ky)+2*cos(kx+ky));
end

69
2020.01.05_calculation_of_Chern_number_by_definition_method/calculation_of_Chern_number_by_definition_method.py Normal file
View File

@@ -0,0 +1,69 @@
"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/3932
"""
import numpy as np
from math import *
import time
def hamiltonian(kx, ky): # 量子反常霍尔QAH模型（该参数对应的陈数为2）
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
matrix = np.zeros((2, 2), dtype=complex)
matrix[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
matrix[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
matrix[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
matrix[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return matrix
def main():
start_time = time.time()
n = 100 # 积分密度
delta = 1e-9 # 求导的偏离量
chern_number = 0 # 陈数初始化
for kx in np.arange(-pi, pi, 2*pi/n):
for ky in np.arange(-pi, pi, 2*pi/n):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
# print(np.argsort(np.real(eigenvalue))[0]) # 排序索引（从小到大）
# print(eigenvalue) # 排序前的本征值
# print(np.sort(np.real(eigenvalue))) # 排序后的本征值（从小到大）
H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
# 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature)
F = (A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta
# 陈数(chern number)
chern_number = chern_number + F*(2*pi/n)**2
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60)
if __name__ == '__main__':
main()

107
2020.01.05_calculation_of_Chern_number_by_definition_method/find_smooth_gauge_and_calculate_Chern_number.py Normal file
View File

@@ -0,0 +1,107 @@
"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/3932
"""
import numpy as np
from math import *
import time
import cmath
def hamiltonian(kx, ky): # 量子反常霍尔QAH模型（该参数对应的陈数为2）
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
matrix = np.zeros((2, 2), dtype=complex)
matrix[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
matrix[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
matrix[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
matrix[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return matrix
def main():
start_time = time.time()
n = 20 # 积分密度
delta = 1e-9 # 求导的偏离量
chern_number = 0 # 陈数初始化
for kx in np.arange(-pi, pi, 2*pi/n):
for ky in np.arange(-pi, pi, 2*pi/n):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
vector_delta_kx = find_vector_with_the_same_gauge(vector_delta_kx, vector)
vector_delta_ky = find_vector_with_the_same_gauge(vector_delta_ky, vector)
vector_delta_kx_ky = find_vector_with_the_same_gauge(vector_delta_kx_ky, vector)
# 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature)
F = (A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta
# 陈数(chern number)
chern_number = chern_number + F*(2*pi/n)**2
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60)
def find_vector_with_the_same_gauge(vector_1, vector_0):
# 寻找近似的同一的规范
phase_1_pre = 0
phase_2_pre = pi
n_test = 10001
for i0 in range(n_test):
test_1 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_1_pre) - vector_0))
test_2 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_2_pre) - vector_0))
if test_1 < 1e-8:
phase = phase_1_pre
# print('Done with i0=', i0)
break
if i0 == n_test-1:
phase = phase_1_pre
print('Gauge Not Found with i0=', i0)
if test_1 < test_2:
if i0 == 0:
phase_1 = phase_1_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else:
phase_1 = phase_1_pre
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else:
if i0 == 0:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre+(phase_2_pre-phase_1_pre)/2
else:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre
phase_1_pre = phase_1
phase_2_pre = phase_2
vector_1 = vector_1*cmath.exp(1j*phase)
# print('二分查找找到的规范=', phase)
return vector_1
if __name__ == '__main__':
main()

90
2020.01.05_calculation_of_Chern_number_by_definition_method/fix_gauge_and_calculate_Chern_number.py Normal file
View File

@@ -0,0 +1,90 @@
"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/3932
"""
import numpy as np
from math import *
import time
import cmath
def hamiltonian(kx, ky): # 量子反常霍尔QAH模型（该参数对应的陈数为2）
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
matrix = np.zeros((2, 2), dtype=complex)
matrix[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
matrix[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
matrix[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
matrix[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return matrix
def main():
start_time = time.time()
n = 20 # 积分密度
delta = 1e-9 # 求导的偏离量
chern_number = 0 # 陈数初始化
for kx in np.arange(-pi, pi, 2*pi/n):
for ky in np.arange(-pi, pi, 2*pi/n):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
index = np.argmax(np.abs(vector))
precision = 0.0001
vector = find_vector_with_fixed_gauge_by_making_one_component_real(vector, precision=precision, index=index)
vector_delta_kx = find_vector_with_fixed_gauge_by_making_one_component_real(vector_delta_kx, precision=precision, index=index)
vector_delta_ky = find_vector_with_fixed_gauge_by_making_one_component_real(vector_delta_ky, precision=precision, index=index)
vector_delta_kx_ky = find_vector_with_fixed_gauge_by_making_one_component_real(vector_delta_kx_ky, precision=precision, index=index)
# 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature)
F = (A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta
# 陈数(chern number)
chern_number = chern_number + F*(2*pi/n)**2
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60)
def find_vector_with_fixed_gauge_by_making_one_component_real(vector, precision=0.005, index=None):
vector = np.array(vector)
if index == None:
index = np.argmax(np.abs(vector))
sign_pre = np.sign(np.imag(vector[index]))
for phase in np.arange(0, 2*np.pi, precision):
sign = np.sign(np.imag(vector[index]*cmath.exp(1j*phase)))
if np.abs(np.imag(vector[index]*cmath.exp(1j*phase))) < 1e-9 or sign == -sign_pre:
break
sign_pre = sign
vector = vector*cmath.exp(1j*phase)
if np.real(vector[index]) < 0:
vector = -vector
return vector
if __name__ == '__main__':
main()